Let $\gamma(t)$ be a regular curve lies on a sphere $S^2$ with center $(0, 0, 0)$ (origin) and radius $r$. Show that the curvature of $\gamma$ is non-zero, i.e., $κ \ne 0$.

Furthermore, if the torsion of the curve $\tau \ne 0$ we have:

$\gamma(t)= - p \overrightarrow{n} - p'\alpha \overrightarrow{b} $

where: $p=1/k, \\ \alpha=1/\tau, \\ r^2=p^2+(p'\alpha)^2 $


This is a standard sort of problem in differential geometry. Assume $\gamma$ is arclength parametrized and write $\gamma = aT+bN+cB$ where $T,N,B$ is the Frenet frame, and $a,b,c$ are smooth. What does the fact that $\gamma$ lies on $S^2$ tell you?


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